Nonlinear Design. Bezier curves

Bezier curves

hroughout centuries, a ruler and a pair of compasses remained
the only practical devices for approximating abstract geometric figures
in real-world drawings and designs. Consequently, straight lines and
circular arcs were the only geometric objects that could be reliably and
accurately reproduced on paper. More complex lines either required
arduous calculations (in mathematical drawings), or were just drawn by
hand (in visual arts and design).

In the 20th century, however, this deficiency proved to be a real
obstacle, especially in industrial design where complex curvilinear
lines and surfaces must be defined precisely so as to be uniformly
reproduced in metal or plastic. When you think of curvilinearity in
technics, it's likely that one of the first things that comes to
mind are the smooth contours of modern cars. Indeed it was the French
Rénault car manufacturing corporation for which Pierre
Bézier developed, in 1960s, a new design tool based on one very
versatile variety of mathematical curves.

These curves, named Bezier curves after their inventor, are now familiar
to any user of a vector drawing program. The most commonly used
Bezier curves of third order are fully defined by four points: two
endpoints and two control points that do not lie on the curve itself but
define its shape (Fig. 1).

Fig. 1:
The anatomy of a Bezier curve

This simple construction is surprisingly versatile. A Bezier curve may
be smooth and streamlined, but it may have very sharp turns and even
fractures; it may have one or two bends, and even form loops
(Fig. 2). It is also very capable at approximating other types of
curves. For example, while you cannot draw an absolutely exact
circle with Bezier curves, you can approximate a quarter of a circle
(i.e. a 90o arc) by a Bezier with an error of less than
0.06%, which is certainly negligible for most practical cases.

Fig. 2:
Exercising a Bezier

Now, Bezier curves are used almost exclusively for creating curvilinear
shapes in all fields of design, from purely technical plans and
blueprints to the most creative artistic genres. Their mathematical
elegance is worth an article of its own; but for now, let's explore
their creative capabilities from a visual, non-mathematical point of view.
In the next sections, we'll try to discover the laws governing the world
of Bezier curves, in the hope that these laws can, at least partially,
explain their beauty and expression and provide directions for
successfully using them in design.